Wednesday, September 24, 2014

Is It The Same Temperature In Every Possible World? Part 3: The Pure Wave Function

EDIT: I uploaded a version that didn't include a major point I wanted to make.

A View of Mount Fuji

In the last post, we talked about how in quantum physics there are new ways for a particle to move. The new ways I called "fluctuating", a particle can fluctuate to you in addition to moving in a more traditional way. I'll call the old method "rolling". This opens up new opportunities for a particle, since a particle that can fluctuate and roll can get into places that a particle that could only roll around couldn't. Take moving around carbon. I can roll you a lump of coal if you aren't too far uphill, but I can burn it and let it waft (fluctuate through the air) over to you no matter how high you are. Getting the coal to you on a large mountain is analogous to getting two protons to fuse. Since the proton can fluctuate, it can get close to another proton even if it doesn't have enough energy to overcome the electric repulsion by direct movement.

Imagine a ball sitting on the bottom of an angle iron. At either end of the iron we have detectors, so that we know that the ball is at one end of the angle iron at \( t_1 \) and at the other end at \( t_2 \). Classically, the ball can only really do one thing, roll down the piece of metal on the base of the joint. This means that the classical view of the particle's position is something like this:

Rolling

But quantumly, the ball can go on paths other than the base. The ball can fluctuate up the side of the angle iron.

Fluctuating

All of these paths are consistent with the experiment/our knowledge of the world/the boundary conditions. Just like the cloud of coal particles that waft toward you, it isn't the case that this quantum particle goes on a particular one of the paths. From the perspective of the starting time, the future path is undetermined, from the perspective of the ending time, the past path is undetermined. But indeterminism isn't a theory, and we're now ready to see what it is that quantum theory does determine. Let's look at the above situation from the top.

The Top

Obviously, these aren't all of the paths that the particle fluctuates on. The drawing is topological. I draw more lines to denote heavily traveled areas, less to indicate regions that the particle rarely sees. Does that sound familiar?

Electric Field

This new mode of movement makes the position of a particle not into a single point, but rather a field of "existence", or to use standard terminology "probability". This field has many of the properties you'd expect a field to have, such as continuity, the existence of a current, etc. This field, for all intents and purposes is the particle. The structure of quantum mechanics allows us to totally determine the field. It is this fact that allows us to determine precisely the predictions of quantum theory. Every particle is a geometry, the structure of the particle determines the shape of that field. For instance, the electric field is the geometry created by the existence of the photon.

The Experiment Below

Experiments can be designed to show that this geometry can't be given by a classical particle. Let's say that the quantum particle is slow, much slower than light. Put a pair of sensors are placed on the angle iron mentioned above. The first sensor is placed in an area that is not accessible by classical movement, but it can fluctuate up there. When activated, it turns on the second sensor placed in an area that is classically acceptable. If the second sensor is ever activated at all, then you know that the particle is moves in a quantum way. Nice and binary.

Now be careful about thinking about this. It is not the case that the particle randomly chooses one of these paths and follows it. It fluctuates out over all of these paths. As it turns out, there are experimental consequences of this. I'll go over this in a couple posts.

Finished With Step 0

Now that I've given the flavor of quantum mechanics, a couple of steps remain before we get into the interpretation matters. The above gives the quantum idea of a single particle, in fact there are multiple particles in the universe. This gives rise to a density matrix and entanglement. More difficultly, the above gives the field that is a single particle, but we know in this world there are also fields. If particles quantize to be fields, what do fields quantize to? Finally, I need to show that the above arguments lead to the Schrodinger equation (there's a standard argument due to Feynman). But that can wait for later. See You Next Time!